scholarly journals Compactness theorems for the Bakry-Emery Ricci tensor on semi-Riemannian manifolds

2017 ◽  
Vol 58 (1) ◽  
pp. 79-86
Author(s):  
 Santos M. S.
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Compactness Theorems

Pseudo B-symmetric manifolds

2017 ◽  
Vol 14 (09) ◽  
pp. 1750119
Author(s):  
Young Jin Suh ◽  
Carlo Alberto Mantica ◽  
Uday Chand De ◽  
Prajjwal Pal
Keyword(s):  
Riemannian Manifold ◽  
Explicit Form ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Sufficient Condition ◽  
Conformally Flat ◽  
Harmonic Curvature ◽  
Codazzi Type ◽  
Curvature Tensors

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

2014 ◽  
Vol 11 (06) ◽  
pp. 1450056 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Higher Dimensions ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Differential Structure ◽  
Arbitrary Signature

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

On weakly conformally symmetric pseudo-Riemannian manifolds

2017 ◽  
Vol 29 (03) ◽  
pp. 1750007
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Integral Curves ◽  
Pontryagin Form ◽  
Lorentzian Case

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the [Formula: see text]-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in [Formula: see text]. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric [Formula: see text]-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

RECURRENT Z FORMS ON RIEMANNIAN AND KAEHLER MANIFOLDS

2012 ◽  
Vol 09 (07) ◽  
pp. 1250059 ◽  
Author(s):  
CARLO ALBERTO MANTICA ◽  
YOUNG JIN SUH
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Conformally Flat ◽  
Kaehler Manifolds ◽  
Closedness Property

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named ( ZRF )n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank (Zkl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for( ZRF )n manifolds with rank (Zkl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric ( PZS )n and weakly Z-symmetric manifolds ( WZS )n in the case of non-singular Z tensor. In the last sections we study special conformally flat ( ZRF )n and give a brief account of Z recurrent forms on Kaehler manifolds.

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